What would be some large (largest?) class of topological spaces for which $\chi$ is defined, so that all standard properties hold, for example that $\chi(X)=\chi(Y)+\chi(Z)$ if $X=Y \cup Z$, ($Y\cap Z=0$).

ADDED. The answer of Algori indicates that a reasonably large class of spaces for which Euler characteristics can be defined are locally compact spaces $X$, whose one point compactification $\bar X$ is a CW complex. Then we can define $\chi(X)=\chi(\bar X)-1$. For example, the Euler characteristics of an open interval according to this definition is $-1$.
This definition rases a second (maybe obvious) question.

Question 2. Suppose $X$ is a locally compact space whose 1 point compactification is a $CW$ complex, and $Y$ is a subspace of $X$ such that both $Y$ and $X\setminus Y$ have this property. Is it ture that $\chi(X)=\chi(Y)+\chi(X\setminus Y)$?

Also, I was thinking, that Euler characteristics is more fundamental then homology.But can it be defined for spaces, where homology is not defined?

Finally, Quiaochu pointed out below that a very similar question was already discussed previously on mathoverflow.

Much like divergent series, there is more than one extension of the Euler characteristic to spaces other than finite CW-complexes. E.g. there is the rational-cohomology Euler characteristic or the Morava K-theory Euler characteristic K(n), which assign 1 and p^n to the classifying space of a cyclic group of order p respectively, or one can apply divergent series techniques to form the alternating sum of dimensions.
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Tyler LawsonMay 29 '10 at 19:40

The open interval is a(n infinite) CW-complex.
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algoriMay 29 '10 at 23:03

2 Answers
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The answer to the question as it is stated is that there is probably no "largest" class of spaces for which the Euler characteristic makes sense.

The answer also depends on where you would like the Euler characteristic to take values. Here is the tautological answer (admittedly not a very exciting one): if you have a category $C$ of spaces closed under taking cones and cylinders, then there is the universal Euler characteristic for that category: just take the free abelian group $K(C)$ that has a generator $[X]$ for each $X\in C$ and quotient it by the span of $[X]+[Cone(f)]-[Y]$ for all $X,Y\in C$ and any morphism $f:X\to Y$ in $C$. The Euler characteristic of any $X$ in $C$ is set to be $[X]$. (There may be variations and/or generalizations of this approach.)

The group $K(C)$ is complicated in general but for some choices of $C$ it has interesting quotients. This can happen e.g. when $C$ admits a good "cohomology-like" functor. For example if $C$ is the category of spaces with finitely generated integral homology groups then $K(C)$ maps to $\mathbf{Z}$ and this gives the usual Euler characteristic. If one takes $C$ to be formed by spaces that admit a finite cover with finitely generated integral homology groups (typical examples are the classifying spaces of $SL_2(\mathbf{Z})$ and more generally of mapping class groups), then $K(C)$ does not map to $\mathbf{Z}$ any more, but it maps to $\mathbf{Q}$. This gives the rational Euler characteristic.

Finally, let me address the last remark by Dmitri. For some categories the group $K(C)$ maps to $\mathbf{Z}$ in several different ways. Let us take e.g. $C$ to be the category formed by spaces whose one-point compactification is a finite CW-complex (with proper maps as morphisms). Then there are (at least) two characteristics; one is obtained using the ordinary cohomology and another one comes from the Borel-Moore homology. On complex algebraic varieties both agree. But the Borel-Moore Euler characteristic of an open $n$-ball is $(-1)^n$.

Here is the answer to the second question: suppose $Y$ is a locally closed subspace of a locally compact space $X$ such that $X,Y,\bar Y,\bar Y\setminus Y, X\setminus\bar Y$ and $X\setminus Y$ are of the form "a finite CW-complex minus a point". Then $\chi(Y)+\chi(X\setminus Y)=\chi(X)$ where $\chi$ is the Euler characteristic computed using the Borel-Moore homology.

The case when $Y$ is closed follows from the Borel-Moore homology long exact sequence. In general we can write $\chi(X)=\chi(X\setminus\bar Y)+\chi(\bar Y)=\chi(X\setminus\bar Y)+\chi(\bar Y\setminus Y)+\chi(Y)$. In the last sum the sum of the first two terms gives $\chi(X\setminus Y)$ since $X\setminus\bar Y$ is open in $X\setminus Y$.

Algori, thanks a lot! Are there some readable references on what you have stated (for example about K(C) and Borel-Moore)?
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DmitriMay 29 '10 at 23:30

Dmitri -- unfortunately I don't have a reference. But to make up for it here are some comments: 1. the definition of $K(C)$ is a slight variation of the definition of the motivic measure, which in turn goes back to other similar definitions (e.g. the Grothendieck group for coherent sheaves etc) 2. the Borel-Moore homology of a space $X$ (with constant coefficients) is the homology of the one point compactification modulo the added point.
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algoriMay 30 '10 at 1:32

The Euler characteristic is the alternate sum of the rank of the integer homology groups of the space.

So how do we end up with an integer ?

The homology groups should be finitely generated. This is the case for finite CW complexes, for spherical complexes (obtained by attaching finitely many cells to a finite set),
for compact subspaces of euclidean spaces that are Absolute Neighborhood retracts (e.g. compact topological manifolds),
and of course for any space homotopy equivalent to such spaces. Note that only finitely many homology groups are non-zeros for these spaces. The Euler characteristic is a well-defined integer and the standard properties hold.

Let me add a few remarks on other interesting kind of spaces.
Some infinite dimensional manifolds have finitely generated homology groups, but infinitely many of them may be non-zero, e.g. $P^\infty C$. In order to extend the Euler characteristic to such spaces, one may want to allow $\infty$ as a possible value for the Euler characteristic (but beware of the $\infty - \infty$ problem), or try to use a normalisation procedure to get some real number (I heard a talk about that but unfortunately I can't recall any reference). I think that it makes sense to say, e.g., that the euler characteristic of $P^\infty C$ is infinite. If we restrict our attention to (inductive limits of) complex manifolds for example, all odd Betti numbers are zero, so one may try to
define an Euler characteristic with values in $N\cup \lbrace \infty\rbrace$ and I would not be surprised if the standard properties hold for such an extension (any counterexamples welcome).

Finally, when leaving the category of finite CW-complexes, I think that the Euler characteristic actually depends on what homology theory you are using. I would be interested in some feedback here. Borel-Moore homology is said to be better behaved than singular homology in that respect.

This is in no way a contradiction or criticism of your excellent answer, but I just want to point out that, w.r.t. "So how do we end up with an integer?"---we don't always end up with an integer. For example, orbifolds have (in general) fractional Euler characteristics.
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Joseph O'RourkeMay 29 '10 at 23:16

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Right; for example, for G a finite group the Eilenberg-Maclane space K(G, 1) should have Euler characteristic 1/|G|. The simplest example of this is when G = Z/2Z, for which K(Z/2Z, 1) is RP^{\infty}. The alternating sum of the Betti numbers is Grandi's series!
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Qiaochu YuanMay 30 '10 at 0:27